Show GL/O/SO(n,R) form groups under Matrix Multiplication

1. The problem statement, all variables and given/known data
Show that the set GL(n, R) of invertible matrices forms a group under matrix multiplication. Show the same for the orthogonal group O(n, R) and the special orthogonal group SO(n, R).

2. Relevant equations

3. The attempt at a solution
So I know the properties that define a group are
- Set with binary operation (in this case matrix multiplication)
- Axiom of Closure (operation produces another element of the set)
- Axiom of Invertibility (There's an inverse (multiplicative and additive))
- Axiom of Identity (There's an identity element (multiplicative and additive))
- Axiom of Associativity (There's an associative property (additive and multiplicative))

I would imagine we need to show that the elements of GL(n,R), O(n,R) and SO(n,R) satisfy all these properties. My question comes in because I'm not able to find lists of any of the elements of these groups, just their descriptive properties. I.e., the Orthogonal group O(n,R) is the group of distance preserving transformations of euclidean n-space and can be represented by (n x n) matrices whose inverse equals their transpose.

So is it possible to show all these properties are satisfied from simply working with the various properties of the groups or would I need to perform calculations on elements of the set to say show its closed under matrix mult., or show what element is the identity/inverse element, or demonstrate the associativity? Any help on how to proceed would be greatly appreciated.

Staff: Mentor

1. The problem statement, all variables and given/known data
Show that the set GL(n, R) of invertible matrices forms a group under matrix multiplication. Show the same for the orthogonal group O(n, R) and the special orthogonal group SO(n, R).

2. Relevant equations

3. The attempt at a solution
So I know the properties that define a group are
- Set with binary operation (in this case matrix multiplication)
- Axiom of Closure (operation produces another element of the set)
- Axiom of Invertibility (There's an inverse (multiplicative and additive))
- Axiom of Identity (There's an identity element (multiplicative and additive))
- Axiom of Associativity (There's an associative property (additive and multiplicative))

I would imagine we need to show that the elements of GL(n,R), O(n,R) and SO(n,R) satisfy all these properties.

Yes.

My question comes in because I'm not able to find lists of any of the elements of these groups, just their descriptive properties. I.e., the Orthogonal group O(n,R) is the group of distance preserving transformations of euclidean n-space and can be represented by (n x n) matrices whose inverse equals their transpose

The Wikipedia entries for these groups or the definition in your book should do.

So is it possible to show all these properties are satisfied from simply working with the various properties of the groups or would I need to perform calculations on elements of the set to say show its closed under matrix mult., or show what element is the identity/inverse element, or demonstrate the associativity? Any help on how to proceed would be greatly appreciated.

This all depends on what you are allowed or willing to use (your empty section 2 of the template!). E.g. invertible matrices have a invertible determinant and the determinant is multiplicative. Similar properties can be used for the other examples.
E.g. associativity holds true for matrix multiplication in general, so it needs only once to be shown: pick a single entry of the product matrix.

Allow me a personal remark: You will probably get more resonance if you formulate your questions shorter and more snappy.

Ahh I see so basically tease out the group axioms by the various properties of those 3 specific groups by their definitions (in say wikipedia or the book). I see you keep mentioning my section 2 as being empty. it's not like I don't know what that section is there for or haven't used it before (look through my previous threads) its just that I wasn't sure what equation I could put down that would help me here or in my other post thats similar to this one. It's not as though we're, say calculating the arc-length or curvature or some parameterized path and you know "Okay I'll need the formulas for these quantities" so don't think I'm neglecting them on purpose, please.

I appreciate your criticism, it's something I've known for a while and I have been trying to minimize my sentences and speak/type less in general. Be more deliberate about my communication but it's a difficult thing to do for me for some reason. Just more room for growth I suppose!